As Paracelsus observed roughly 500 years ago, "the dosage is the poison".
There is a small, but generally positive, body of research, on the use of oregano (and somewhat less on the use of thyme) or its oil in chicken feed for reducing average bacterial load in the short term. There is, to the best of my knowledge, no long term study. It is entirely possible that routine dosing of a given amount of the active ingredients contained in oregano (or thyme) will simply select for bacteria and other parasites best able to tolerate it, just as acidifying your water via ACV simply selects for differing bacteria in your chicken's gut and has declining effectiveness over time (as well as being highly dependent upon the initial pH of your water - much like shocking a pool with chlorine).
The problem with feeding natural oregano (or most OTC oregano oils) is that you have absolutely no idea what the concentration is of the active ingredients, and therefore can not dose effectively or consistently. Its magical thinking to believe otherwise. That said, in reasonable quantities, there is no effective downside to your chickens eating oregano, or thyme, or a host of other plants should they desire to do so. My flock routinely snatches leaves of my oregano plant from my raised bed where it pokes through the fencing. Does it help??? There is a remote possibility that it might. Does it hurt? Nope.
Do I count on it as part of an effective parasite/bateria/etc control system??? Not a snowball's chance in the Phlegethon. [that's the "River of Fire" in Hades, for those of you not familiar]
I am unfamiliar with any studies as to how much oregano a chicken can consume before it begins to affect the flavor of the egg. "Taste" studies are notoriously unreliable, and highly personalized to both palatte and cultural culinary experience.